`((3-2x)/x^3)/(2/x^2-1/(x^3+x^2))` Simplify the complex fraction.

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To simplify the given complex fraction `((3-2x)/x^3)/(2/x^2-1/(x^3+x^2))` , we may look for the LCD or least common denominator.

The denominators are `x^3` , `x^2`, and`x^3+x^2` .

Note: The factored form of `x^3+x^2 = x^2(x+1)`.

LCD is the same as getting LCM from the denominators.

We get the product of each factor with highest exponent value,

`LCD=x^3*(x+1)` .

Multiply each term by the `LCD=x^3*(x+1).`

`((3-2x)/x^3*x^3*(x+1))/(2/x^2*x^3*(x+1)-1/(x^3+x^2)x^3*(x+1)) `

`((3-2x)/x^3*x^3*(x+1))/(2/x^2*x^3*(x+1)-1/(x^2(x+1))x^3*(x+1)) `

`((3-2x)(x+1))/(2x*(x+1)-1*x)`

`(3x+3-2x^2-2x)/((2x^2+2x)-x)`

`(-2x^2+3x-2x+3)/(2x^2+2x-x)`

`(-2x^2+x+3)/(2x^2+x)`

Final answer:

`((3-2x)/x^3)/(2/x^2-1/(x^3+x^2))=(-2x^2+x+3)/(2x^2+x)`

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